Optimal. Leaf size=15 \[ -\frac {1}{3} \sin ^{-1}\left (\frac {1}{2} \cos (7+3 x)\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.02, antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {3265, 222}
\begin {gather*} -\frac {1}{3} \text {ArcSin}\left (\frac {1}{2} \cos (3 x+7)\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 222
Rule 3265
Rubi steps
\begin {align*} \int \frac {\sin (7+3 x)}{\sqrt {3+\sin ^2(7+3 x)}} \, dx &=-\left (\frac {1}{3} \text {Subst}\left (\int \frac {1}{\sqrt {4-x^2}} \, dx,x,\cos (7+3 x)\right )\right )\\ &=-\frac {1}{3} \sin ^{-1}\left (\frac {1}{2} \cos (7+3 x)\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] Result contains complex when optimal does not.
time = 0.07, size = 39, normalized size = 2.60 \begin {gather*} \frac {1}{3} i \log \left (i \sqrt {2} \cos (7+3 x)+\sqrt {7-\cos (2 (7+3 x))}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(56\) vs.
\(2(11)=22\).
time = 5.33, size = 57, normalized size = 3.80
method | result | size |
default | \(-\frac {\sqrt {\left (3+\sin ^{2}\left (7+3 x \right )\right ) \left (\cos ^{2}\left (7+3 x \right )\right )}\, \arcsin \left (-1+\frac {\left (\cos ^{2}\left (7+3 x \right )\right )}{2}\right )}{6 \cos \left (7+3 x \right ) \sqrt {3+\sin ^{2}\left (7+3 x \right )}}\) | \(57\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.52, size = 11, normalized size = 0.73 \begin {gather*} -\frac {1}{3} \, \arcsin \left (\frac {1}{2} \, \cos \left (3 \, x + 7\right )\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 94 vs.
\(2 (11) = 22\).
time = 0.38, size = 94, normalized size = 6.27 \begin {gather*} \frac {1}{6} \, \arctan \left (-\frac {4 \, \cos \left (3 \, x + 7\right ) \sin \left (3 \, x + 7\right ) - {\left (\cos \left (3 \, x + 7\right )^{3} - 2 \, \cos \left (3 \, x + 7\right )\right )} \sqrt {-\cos \left (3 \, x + 7\right )^{2} + 4}}{\cos \left (3 \, x + 7\right )^{4} - 8 \, \cos \left (3 \, x + 7\right )^{2} + 4}\right ) - \frac {1}{6} \, \arctan \left (\frac {\sin \left (3 \, x + 7\right )}{\cos \left (3 \, x + 7\right )}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sin {\left (3 x + 7 \right )}}{\sqrt {\sin ^{2}{\left (3 x + 7 \right )} + 3}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 48 vs.
\(2 (11) = 22\).
time = 0.78, size = 48, normalized size = 3.20 \begin {gather*} \frac {2}{3} \, \arctan \left (-\frac {1}{2} \, \sqrt {3} \tan \left (\frac {3}{2} \, x + \frac {7}{2}\right )^{2} - \frac {1}{2} \, \sqrt {3} + \frac {1}{2} \, \sqrt {3 \, \tan \left (\frac {3}{2} \, x + \frac {7}{2}\right )^{4} + 10 \, \tan \left (\frac {3}{2} \, x + \frac {7}{2}\right )^{2} + 3}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.07 \begin {gather*} \int \frac {\sin \left (3\,x+7\right )}{\sqrt {{\sin \left (3\,x+7\right )}^2+3}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________